Narrow Results

In 1983 Conway and Gordon proved that any embedding of the complete graph K₇ into ℝ³ contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper, we are interested in knotted Hamiltonian cycles in linear embedding of K₇. Concretely it is shown that any linear embedding of K₇ contains at most three figure-8 knots.

We use the theory of oriented matroids to show that any linear embedding of K₉, the complete graph on nine vertices, into 3-space contains a non-split link with three components. This shows that Sachs' conjecture on linear, linkless embeddings of graphs, whether true or false, does not extend to 3-links.

We show there exists a linear embedding of K_3,3,1 with n nontrivial 2-component links if and only if n = 1, 2, 3, 4, or 5.